The present invention relates generally to coding techniques for communications and, more particularly, to code designing for repeat-zigzag Hadamard codes.
For communication systems operating in the low signal-to-noise ratio (SNR) regime (e.g., code-spread communication systems and power-limited sensor networks), low-rate coding schemes play a critical role. Traditional low-rate channel coding schemes include Hadamard codes and super-orthogonal convolutional codes. These codes offer low coding gain and hence their performance is far away from the ultimate Shannon limit. The ultimate Shannon capacity of an Additive White Gaussian Noise (AWGN) channel in terms of SNR per information bit is about −1.6 dB for codes with rates approaching zero. Turbo-code-based, low-rate concatenated coding schemes have been proposed which offer higher performance but they also incur higher complexities due to the complex trellis structures that specify the codes. The low-density parity-check (LDPC) codes and the repeat-accumulate (RA) codes offer capacity-approaching capability for various code rates when the ensemble profiles are optimized. However, in the low-rate region, they both suffer from significant performance loss and extremely slow convergence with iterative decoding. On the other hand, Hadamard codes have been shown to be a useful tool for improving the code performance in the low-rate region. Constructed from Hadamard code arrays, low-rate turbo-Hadamard codes offer a bit-error-rate (BER) of 10−5 at a signal to noise ratio Eb/N0=−1.2 dB, which is only around 0.4 dB away from the Shannon limit. More recently, built on zigzag codes, parallel-concatenated zigzag-Hadamard (PCZH) codes and repeat-zigzag-Hadamard (RZH) codes have been proposed, both of which exhibit much simpler encoder and decoder structures while still offering a similar performance as that of the turbo-Hadamard codes.
Similar to LDPC codes and RA codes, an RZH code can be represented by its Tanner graph, based on which iterative decoding algorithms can be derived. In two decoding schemes, namely the turbo-like serial decoding and the LDPC-like parallel decoding, it is clear that the parameters of an RZH code need to be chosen carefully for capacity-approaching performance, which leads to the irregular RZH (IRZH) code with optimized degree profile. The exact density evolution method for LDPC code design can be employed for the IRZH code optimization. However, the complexity is extremely high, even for the moderate rate LDPC codes. Density evolution with Gaussian approximation does not work for the design of low-rate RZH codes. Accordingly, there is a need for a design method for low-rate RZH codes.